α-Clustering in atomic nuclei from first principles with statistical learning and the Hoyle state character

A long-standing crucial question with atomic nuclei is whether or not α clustering occurs there. An α particle (helium-4 nucleus) comprises two protons and two neutrons, and may be the building block of some nuclei. This is a very beautiful and fascinating idea, and is indeed plausible because the α particle is particularly stable with a large binding energy. However, direct experimental evidence has never been provided. Here, we show whether and how α(-like) objects emerge in atomic nuclei, by means of state-of-the-art quantum many-body simulations formulated from first principles, utilizing supercomputers including K/Fugaku. The obtained physical quantities exhibit agreement with experimental data. The appearance and variation of the α clustering are shown by utilizing density profiles for the nuclei beryllium-8, -10 and carbon-12. With additional insight by statistical learning, an unexpected crossover picture is presented for the Hoyle state, a critical gateway to the birth of life.


Supplementary
displays the same panels as those in Figure 6, in the form of the two-dimensional presentation. The legend (color code) is the same as the one for Figure 6. The overlap probabilities of the regional Qaligned states (see Figure 6 f-i) are also indicated. Panel j is added in Supplementary Figure 1 compared to Figure  6.
From this two-dimensional figure, the distance between density peaks and the angles among three density peaks can be seen more precisely than from Figure 6, although the basic features of the density distribution can be seen more clearly by the three-dimensional presentation in Figure 6.

Supplementary Note 2. Nucleon-Nucleon Interactions
JISP16 interaction 1 : The nomenclature of this N N interaction comes from the J-matrix Inverse Scattering Potential tuned up to 16 O (JISP16). The JISP16 interaction is a nonlocal N N potential, aiming to minimize the 3N -force effects on ab initio nuclear structure calculations. No explicit three-nucleon terms are included in the JISP16 interaction, but momentum-dependent N N interaction terms produce similar effects. This potential is constructed by using a unitary transformation, the socalled phase-equivalent transformation, in order to tailor the original N N interaction, keeping two-body physical observables unchanged. Through its construction, this interaction fits not only the two-nucleon scattering data and deuteron properties, but also approximately fits the binding energies of selected nuclei up to 16 O.
Daejeon16 interaction 2 : The Daejeon16 N N interaction is a successor of the JISP16 N N interaction. This * otsuka@phys.s.u-tokyo.ac.jp interaction has recently been developed and applied to ab initio nuclear structure calculations in light nuclei. The way to construct the Daejeon16 interaction bears similarities with JISP16, using the phase-equivalent transformation. One of the main differences between these two interactions is the initial Hamiltonian to be evolved via this transformation. The initial interaction of the Dae-jeon16 is taken from the similarity renormalization group (SRG)-evolved chiral effective field theory (χEFT) N N interaction up to the next-to-next-to-next leading order (N3LO), while that of the JISP16 is a much simpler one, the inverse scattering tridiagonal potential.
The lattice calculation 6,7 and their extension to the reaction 8 were performed with a chiral EFT interaction up to N2LO with modifications to the lattice calculations.

Supplementary Note 3. Configuration Interaction (CI) Calculation or Shell-Model Calculation
The shell-model calculation is one of the standard methods in the nuclear many-body problem. It is similar to the Configuration Interaction (CI) calculation in other fields of science. The single-particle orbits are defined first. Protons and neutrons are put into these orbits. They are called valence protons or neutrons. Slater determinants are composed of single-particle states of these valence nucleons. We can construct the Hilbert space spanned by such Slater determinants. Each matrix element of the Hamiltonian is calculated for bra and ket vectors being Slater determinants. Once all matrix elements are calculated, the matrix is diagonalized, to solve the many-nucleon Schrödinger equation. We then obtain energy eigenvalues of this Hamiltonian as well as eigenwavefunctions, from which we can calculate various physical quantities. This is an outline of the conventional Supplementary Figure 1 | Two dimensional presentations of the density profiles on the yz plane of α and 12 C nuclei. All panels correspond to the panels in Figure 6. a Color code of the density. b Density of the α-particle ground state. c-e Density of 0 + states of 12 C nucleus. f-j Decomposition into the regions. The probability in the indicated region is shown.
shell-model calculation. As an example of the conventional shell-model approach, the excitation level energies of 12 C nucleus can be calculated with some phenomenologically fitted N N interactions for small model spaces. For instance, the Cohen-Kurath interaction for the pshell 9 can reproduce excitation energies of many states of so-called p-shell nuclei, but the Hoyle state was out of reach being more than 4 MeV away 9 .
The number of Slater determinants needed in the shellmodel calculation is called the shell-model dimension, and is crucial for the feasibility of actual computation. The maximum dimension, for which the conventional shell-model calculation can be performed, is about 10 11 at present 10 .
The state-of-the-art approach extending the ordinary shell model is the ab initio No-Core Full Configuration (NCFC). This was proposed for the calculations with ab initio N N interactions and has been applied to the structure of p-shell nuclei, producing salient descriptions of these nuclei (see reviews 11,12 ). As an example, the ro- tational bands were studied for Be isotopes, etc. 13,14 as mentioned in Results. NCFC calculations produce energy eigenvalues and E2 properties of 12 C consistent with the present work, for instance 15 . An NCFC calculation on 9 Be with the JISP16 interaction shows the density profile in the laboratory frame for the J π =3/2 − ground state of the 9 Be nucleus 16 . This density profile exhibits two peaks of the proton density apart from each other by 2 fm, whereas the peaks are about 3.5 fm away from each other in Figure 3d. The sharply peaked density profile in the snapshot state (or intrinsic state in literatures), like the one in Figure 3d, is smeared out in the density profile in the laboratory frame because of the couplings of angular momenta (see Figure 5 of ref. 16 ). Although the excess neutron in 9 Be somewhat aligns the ground state, the laboratory-frame density profile, calculable directly from the shell-model wave function, is still far from the snapshot we find for 8 Be, or for 10 Be, that establishes clustering. Thus, this case with 9 Be provides another example of the crucial role of the snapshot (or intrinsic) state, as emphasized in Results.
In order to avoid the afore-mentioned difficulty of the shell model, the symmetry-adapted no-core shell-model (SA-NCSM) and the no-core symplectic model (NCSpM) have been proposed (see recent review 17 ) and applied: the former is performed with ab initio interactions 18,19 , while the latter is with empirical interactions 20 . The numerical calculation was made possible by truncating the many-body Hilbert space into its particular subsets with designated symmetries 21 .
Another approach to overcome the difficulty of exploding shell-model dimension is the Monte Carlo Shell Model (MCSM) [22][23][24][25] , which is explained in Methods. We note that the no-core MCSM was applied earlier to Be isotopes with more limited single-particle degrees of freedom and a different N N interaction 26 .
The MCSM also has the advantage of providing a way to visualize the shape of each MCSM eigenstate through what is called the T-plot 27,28 . Because the MCSM basis vector is a deformed Slater determinant, one can calculate its intrinsic quadrupole moments, i.e. the quadrupole moments in the body-fixed frame. They can be expressed by two parameters β 2 and γ, as described in Results. The importance of each MCSM basis vector to a given eigenstate (its overlap probability in the MCSM eigenstate) is represented by the size (area) of its circular symbol in the T-plot. The T-plot is made on the PES, and intuitively exhibits the underlying physical pictures for the states of interest as demonstrated in a variety of studies, e.g. in Refs. [29][30][31][32][33][34][35] .

Supplementary Note 4. Other Theoretical
Approaches to the α Clustering A shell-model approach has been reported in a different context from the viewpoint of the so-called cluster-shell competition 36,37 : the mixing of conventional shell-model states with α clustering states was discussed by using empirical N N interactions, leading to a different picture compared to the present work. No discussion was reported on triangular configurations in these works 36,37 . The α cluster structure has been considered also in other theoretical models, including the Generator Coordinator Method 38-40 , the Fermion Molecular Dynamics 41 , the Antisymmetrized Molecular Dynamics 42 , Bose-Einstein condensate 43,44 , Relativistic Mean Field 45 and the lattice-simulation 6,7 frameworks, where the Pauli principle among nucleons is activated while the wave functions are constrained dynamically.
The triangular configuration appears as a key feature in the present work, similar to some other works, e.g. 6 . But the actual triangles seem to be rather complex and fluctuate as shown also in the present work. The relationship of our results to simple models, e.g. equilateral triangles 46,47 , is an open and interesting question.

Supplementary Note 5. Some Observables of 12 C
The calculated value of B(E2; 0 + 2 → 2 + 1 ) of 12 C is somewhat smaller than the quoted experimental value. The experimental value may be changed to a larger value (personal communications from Kibédi, T., Stuchbery, A. E. and Gorgön, A. based on the data 48 ). The present calculated value is larger than some other theoretical values (see, for instance, a review 44 ), but turns out not to be large enough. If more single-particle orbits are included as stated above, the nucleus may be somewhat more deformed or may have somewhat more developed clustering, and thereby the present calculated value may increase. This point motivates larger calculations in the future. In such future calculations, one expects the excitation energy of the 0 + 2 (Hoyle) state to further decrease towards the experimental one.
Likewise, the radius and E0 transition of the Hoyle state can be improved if more single-particle orbits are included. However, the ground state properties will likely be less influenced, as expected from Supplementary Figure 3. The transition densities of elastic and inelastic scatterings are of interest, and will be analyzed including the finite-size effects of proton and neutron.